In this paper we develop a numerical scheme for approximating interior jump discontinuity solutions of compressible Stokes flows with inflow jump datum. The scheme is based on a decomposition of the velocity vector into three parts: the jump part, an auxiliary one and the smoother one. The jump discontinuity is handled by constructing a vector function extending the density jump value of the normal vector on the interface to the whole domain. We show existence of the finite element solutions for the three parts, derive error estimates and also convergence rates based on the piecewise regularities. Numerical examples are given, confirming the derived convergence rates.In this paper we develop a numerical scheme for approximating interior jump discontinuity solutions of compressible Stokes flows with inflow jump datum. The scheme is based on a decomposition of the velocity vector into three parts: the jump part, an auxiliary one and the smoother one. The jump discontinuity is handled by constructing a vector function extending the density jump value of the normal vector on the interface to the whole domain. We show existence of the finite element solutions for the three parts, derive error estimates and also convergence rates based on the piecewise regularities. Numerical examples are given, confirming the derived convergence rates.